Presenting Schur algebras as quotients of the universal enveloping algebra of 𝔤𝔩_{2}

Stephen Doty and Anthony Giaquinto

October 26, 2000

Abstract

We give a presentation of the Schur algebras S_{ℚ}(2, d) by generators and relations, in fact a presentation which is compatible with Serre’s presentation of the universal enveloping algebra of a simple Lie algebra. In the process we find a new basis for S_{ℚ}(2, d), a truncated form of the usual PBW basis. We also locate the integral Schur algebra within the presented algebra as the analogue of Kostant’s ℤ-form, and show that it has an integral basis which is a truncated version of Kostant’s basis.

1 Introduction

Consider a 2-dimensional vector space E over the rational field ℚ, which shall remain fixed throughout the paper, and fix a basis e_{1}, e_{2} for E. Via this basis for E the group of linear automorphisms of E will be identified with the group GL_{2}(ℚ) and the vector space End(E) under Lie bracket [x, y] = xy−yx will be identified with the Lie algebra 𝔤𝔩_{2}(ℚ). The action of the group on E extends diagonally to an action on tensor space E^{⊗d}. The corresponding representation

σ_{d} : GL_{2}(ℚ) → End(E^{⊗d})

extends by linearity to a representation

σ_{d} : ℚGL_{2}(ℚ) → End(E^{⊗d})

of the group algebra ℚGL_{2}(ℚ), and the Schur algebra S(2, d) is precisely the image of the representation. By differentiating σ_{d} we obtain a representation ρ_{d} of the Lie algebra 𝔤𝔩_{2}(ℚ) which extends linearly to a representation

ρ_{d} : U(𝔤𝔩_{2}) → End(E^{⊗d})

of the universal enveloping algebra, and its image is the same as the image of the group algebra. Thus S(2, d) is a homomorphic image of U(𝔤𝔩_{2}).

Now if the original representation σ_{d} is restricted to the subgroup SL_{2}(ℚ) it is obvious that the image of its group algebra in End(E^{⊗d}) will still be the same as the image of the group algebra of GL_{2}(ℚ) since the two groups differ only in scalars and End(E^{⊗d}) already contains all the scalars. So the restriction of ρ_{d} gives a representation of 𝔰𝔩_{2} and it follows that S(2, d) is also a homomorphic image of U(𝔰𝔩_{2}).

Serre has given a presentation of U(𝔤) for any semisimple Lie algebra 𝔤 and, in particular, for U(𝔰𝔩_{2}). This is easily adapted to give a presentation of U(𝔤𝔩_{2}) (see §3). Thus the natural question arises: to find an efficient set of generators of the kernel of ρ_{d} in terms of the Lie algebra generators of U(𝔰𝔩_{2}) or U(𝔤𝔩_{2}). In other words, what additional relations must be imposed on the generators of the universal enveloping algebra in order to define a presentation of the Schur algebra? This is the same as the question of finding a set of generators for the annihilator of the U-module E^{⊗d} (U = U(𝔤𝔩_{2}) or U(𝔰𝔩_{2})).

In this paper we obtain a precise answer to this question. Write e_{ij} for the usual matrix units, defined in terms of Kronecker’s delta by e_{ij} = (δ_{il}δ_{jl})_{i,j}. In the Lie algebra 𝔤𝔩_{2}, set e = e_{12}, f = e_{21}, H_{i} = e_{ii} for i = 1, 2, and h = H_{1} − H_{2}. One can easily compute the eigenvalues of the images under the above representation of H_{i} and h in End(E^{⊗d}) and thereby determine the minimal polynomial of those endomorphisms. Then our result is that the relation given by the minimal polynomial is precisely the additional relation needed to describe the desired presentation of S(2, d). In other words, the kernel of the quotient map U(𝔰𝔩_{2}) → S(2, d) is generated by the minimal polynomial of the image of h. Similarly, the kernel of the quotient map U(𝔤𝔩_{2}) → End(E^{⊗d}) is generated by the minimal polynomials of the images of H_{1} and H_{2}, along with one additional relation, H_{1} + H_{2} = d, which can be used to eliminate one of the two generators H_{i} from the presentation. In proving these results we obtain along the way a new basis for S_{ℚ}(2, d), which is a truncated form of the usual PBW basis of the universal enveloping algebra.

The algebra S(2, d) = S_{ℚ}(2, d) contains a ℤ-order S_{ℤ}(2, d), the integral Schur algebra, and if k is any field we have an isomophism S_{k}(2, d) ≃ S_{ℤ}(2, d)⊗_{ℤ}k. (See [Gr].) We show that the integral Schur algebra is precisely the analogue of Kostant’s ℤ-form in S_{ℚ}(2, d), i.e. it is the subring generated by all divided powers of e, f. Moreover, it has an integral basis which is precisely a truncated version of the usual basis of the Kostant ℤ-form. (This basis is closely related to the work of Richard Green [RG].)

Although we obtain these results over the rational field ℚ all the arguments can be carried out equally well over any field of characteristic zero, so the same presentation holds over any such field, and, in particular, over the complex field ℂ.

For many problems in Lie theory, the rank 1 case turns out to be of fundamental importance for the general case, and that is why we devote an entire paper just to that case. We expect to treat the general case in a later paper, in which one cannot expect to find such explicit reduction formulas as those given here. The quantized version of these results is addressed in [DG], where different techniques of proof are developed.

2 Statement of results

We now describe our results more precisely.

Our first main result describes the Schur algebra S(2, d) over the rational field ℚ in terms of generators and relations. The result is pleasant: S(2, d) has the same presentation as U(𝔰𝔩_{2}) with just one additional relation.

2.1 Theorem Over ℚ, the Schur algebra S(2, d) is isomorphic with the associative algebra (with 1) generated by e, f, h subject to the relations:

he − eh = 2e;  ef − fe = h;  hf − fh = −2f; (a)

(h + d)(h + d − 2) ⋯ (h − d + 2)(h − d) = 0. (b)

Moreover, this algebra has a “truncated PBW” basis over ℚ consisting of all f^{a}h^{b}e^{c} such that a + b + c ≤ d.

Note that taking d → ∞ in the above recovers the usual presentation of U(𝔰𝔩_{2}). Moreover, it follows from the above that for each d there is a surjective quotient mapping S(2, d + 2) → S(2, d) defined by mapping generators onto generators.

By a linear change of variable (h = 2H_{1}−d or h = d−2H_{2}) we obtain from the above theorem two equivalent presentations, which are more convenient for describing the integral Schur algebra.

2.2 Theorem Over ℚ, the Schur algebra S(2, d) is isomorphic with the associative algebra (with 1) generated by e, f, H_{1} subject to the relations:

H_{1}e − eH_{1} = e;  ef − fe = 2H_{1} − d;  H_{1}f − fH_{1} = −f; (a)

H_{1}(H_{1} − 1) ⋯ (H_{1} − d) = 0. (b)

Moreover, this algebra has a “truncated PBW” basis over ℚ consisting of all e^{a}H_{1}^{b}f^{c} such that a + b + c ≤ d.

2.3 Theorem Over ℚ, the Schur algebra S(2, d) is isomorphic with the associative algebra (with 1) generated by e, f, H_{2} subject to the relations:

H_{2}e − eH_{2} = −e;  ef − fe = d − 2H_{2};  H_{2}f − fH_{2} = f; (a)

H_{2}(H_{2} − 1) ⋯ (H_{2} − d) = 0. (b)

Moreover, this algebra has a “truncated PBW” basis over ℚ consisting of all f^{a}H_{2}^{b}e^{c} such that a + b + c ≤ d.

Of course Theorems 2.2 and 2.3 are equivalent to one another via the relation H_{1} + H_{2} = d.

Our next result constructs the integral Schur algebra S_{ℤ}(2, d) in terms of the generators given above.

2.4 Theorem The integral Schur algebra S_{ℤ}(2, d) is isomorphic with the subalgebra of S_{ℚ}(2, d) generated by all divided powers

e^{(m)} := e^{m}/m!; f^{(m)} := f^{m}/m!.

Moreover, this algebra has a “truncated Kostant” basis over ℤ consisting of all

f^{(a)}\binom{H_{2}}{b}e^{(c)} (a + b + c ≤ d) (a)

and another such basis consisting of all

e^{(a)}\binom{H_{1}}{b}f^{(c)} (a + b + c ≤ d). (b)

It would appear to be useful to express the product of two basis elements as a ℤ-linear combination of basis elements. The next result, combined with standard commutation identities in the enveloping algebra, enables one to do this.

2.5 Theorem In S_{ℤ}(2, d) we have the following reduction formulas for any nonnegative integers a, b, c:

(a) f^{(a)}\binom{H_{2}}{b}e^{(c)} = ∑_{k=s}^{min(a,c)}(−1)^{k−s}\binom{k − 1}{s − 1}\binom{b + k}{k}f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k)}

(b) e^{(a)}\binom{H_{1}}{b}f^{(c)} = ∑_{k=s}^{min(a,c)}(−1)^{k−s}\binom{k − 1}{s − 1}\binom{b + k}{k}e^{(a−k)}\binom{H_{1}}{b + k}f^{(c−k)}

where s = a + b + c − d.

3 Enveloping algebras

In this section we fix some basic notation and quote some standard results from the theory of enveloping algebras.

3.1 We write

e = [\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}] f = [\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}] H_{1} = [\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}] H_{2} = [\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}]

for the canonical basis elements of the Lie algebra 𝔤𝔩_{2}. The elements e, f, h := H_{1} − H_{2} form the canonical basis of the Lie subalgebra 𝔰𝔩_{2}.

3.2 The universal enveloping algebra U(𝔤𝔩_{2}) is the associative algebra (with 1) generated by e, f, H_{1}, H_{2} subject to the relations

(a)  H_{1}H_{2} = H_{2}H_{1},
(b)  H_{1}e − eH_{1} = e, H_{1}f − fH_{1} = −f,
(c)  H_{2}e − eH_{2} = −e, H_{2}f − fH_{2} = f,
(d)  ef − fe = H_{1} − H_{2}.

Moreover, U(𝔰𝔩_{2}), the associative algebra (with 1) generated by e, f, h and satisfying the relations

he − eh = e, hf − fh = −f, ef − fe = h, (e)

is isomorphic with the subalgebra of U(𝔤𝔩_{2}) generated by e, f, h = H_{1} − H_{2}. (This follows at once from the Poincare-Birkhoff-Witt theorem.)

3.3 From the Poincare-Birkhoff-Witt theorem it also follows that the algebra U(𝔰𝔩_{2}) (resp., U(𝔤𝔩_{2})) has a ℚ-basis (the PBW-basis) consisting of all

f^{a}h^{b}e^{c} (resp., f^{a}H_{1}^{b_{1}}H_{2}^{b_{2}}e^{c})

as a, b, b_{1}, b_{2}, c range over the nonnegative integers.

3.4 For any nonnegative integer m and any element T of an associative ℚ-algebra with 1 we define

T^{(m)} = T^{m}/(m!), \binom{T}{m} = T (T − 1) ⋯ (T − m + 1)/(m!)

and we define these for negative m to have the value 0. Note that the usual identity

\binom{T}{m + 1} = \binom{T − 1}{m + 1} + \binom{T − 1}{m} (a)

holds true in this situation.

We consider the ℤ-group G_{ℤ} = SL_{2} (resp., GL_{2}) and denote by Dist(G_{ℤ}) its algebra of distributions (see [Ja, Part II, (1.12)]). Dist(G_{ℤ}) is the subring of U(𝔰𝔩_{2}) (resp., U(𝔤𝔩_{2})) generated by all

f^{(m)}, e^{(m)} (resp., f^{(m)}, e^{(m)}, \binom{H_{1}}{m}, \binom{H_{2}}{m}) . (b)

In the case G_{ℤ} = SL_{2} the algebra of distributions coincides with Kostant’s ℤ-form U_{ℤ}(𝔰𝔩_{2}) in U(𝔰𝔩_{2}) and has a ℤ-basis consisting of all

f^{(a)}\binom{h}{b}e^{(c)} (c)

as a, b, c range over the nonnegative integers. In the case G_{ℤ} = GL_{2} the algebra of distributions has a ℤ-basis consisting of all

f^{(a)}\binom{H_{1}}{b_{1}}\binom{H_{2}}{b_{2}}e^{(c)} (d)

as a, b_{1}, b_{2}, c range over the nonnegative integers. We call elements of this form monomials of degree a + b_{1} + b_{2} + c and height a + c. Similar language will be applied to elements obtained from elements of the above form by interchanging H_{1} with H_{2} and e with f.

The following lemma is well-known (see [Ko]) and will be needed later on.

3.5 Lemma Given any polynomial P (t) in ℚ[t] and any nonnegative integer k, we have the following identities in U(𝔤𝔩_{2}):

e P (H_{1}) = P (H_{1} − 1) e, e P (H_{2}) = P (H_{2} + 1) e, (a) f P (H_{1}) = P (H_{1} + 1) f, f P (H_{2}) = P (H_{2} − 1) f, (b)

ef^{(k)} = f^{(k)}e + f^{(k−1)}(H_{1} − H_{2} − k + 1) (c)

e^{(k)}f = fe^{(k)} + (H_{1} − H_{2} − k + 1)e^{(k−1)} (d)

fe^{(k)} = e^{(k)}f + e^{(k−1)}(H_{2} − H_{1} − k + 1) (e)

f^{(k)}e = ef^{(k)} + (H_{2} − H_{1} − k + 1)f^{(k−1)}. (f)

Proof. By linearity it suffices to prove the identities (a) and (b) for P of the form t^{m} where m is a nonnegative integer. But this follows for positive m from 3.2(b),(c) by induction on m, and for m = 0 it is vacuous. The next two identities follow from the relation 3.2(d) by induction on k. Finally, the last two relations follow by symmetry, by interchanging e with f and H_{1} with H_{2}. □

4 The algebra B_{d}

In this section we define an algebra B_{d} (over ℚ) by generators and relations. Eventually B_{d} will turn out to be isomorphic with the Schur algebra S(2, d).

4.1 Fix a nonnegative integer d. We define B_{d} to be the algebra with 1 generated by e, f, H_{1}, H_{2} subject to the relations 3.2(a)–(d), together with the additional relations

H_{1} + H_{2} = d; (a) 0 = H_{1}(H_{1} − 1) ⋯ (H_{1} − d). (b)

Thus by definition B_{d} is a homomorphic image of U(𝔤𝔩_{2}).

Note that in the presence of the relation (a) the relation (b) can be replaced by the equivalent relation

0 = H_{2}(H_{2} − 1) ⋯ (H_{2} − d). (c)

An important remark that will be useful in the sequel is that the defining relations for B_{d} are invariant under the operation of interchanging e with f and H_{1} with H_{2}. We shall refer to this property as symmetry. (This operation defines a Lie algebra automorphism of 𝔤𝔩_{2}, which induces a corresponding automorphism of B_{d}.)

4.2 It follows from the defining relations that in the algebra B_{d} we have

\binom{H_{1}}{d + 1} = 0 = \binom{H_{2}}{d + 1}. (a)

More generally, we have

0 = \binom{H_{1}}{b_{1}}\binom{H_{2}}{b_{2}} (b_{1} + b_{2} ≥ d + 1). (b)

One derives this first in the case b_{1} + b_{2} = d + 1 from 4.1(a) and 4.1(b); the statement for b_{1} + b_{2} > d + 1 then follows immediately.

4.3 Lemma In the algebra B_{d} we have

f^{a}\binom{H_{2}}{b} = 0 = \binom{H_{2}}{b}e^{a}, e^{a}\binom{H_{1}}{b} = 0 = \binom{H_{1}}{b}f^{a}

for any pair a, b of nonnegative integers satisfying a+b = d+1. In particular, e^{d+1} = 0 = f^{d+1}.

Proof. Set b = d + 1 − a and proceed by induction on a. The case a = 0

is true by equations 4.2(a). Supposing by induction that f^{a}\binom{H_{2}}{b} = 0 =

\binom{H_{2}}{b}e^{a}, we obtain

0 = f^{a}\binom{H_{2}}{b} = f^{a}\binom{H_{2}}{b}f

= f^{a+1}\binom{H_{2} + 1}{b} (by Lemma 3.5)

= f^{a+1}\binom{H_{2}}{b} + f^{a+1}\binom{H_{2}}{b − 1}

= f^{a+1}\binom{H_{2}}{b − 1}

and

0 = \binom{H_{2}}{b}e^{a} = e\binom{H_{2}}{b}e^{a}

= \binom{H_{2} + 1}{b}e^{a+1} (by Lemma 3.5)

= \binom{H_{2}}{b}e^{a+1} + \binom{H_{2}}{b − 1}e^{a+1}

= \binom{H_{2}}{b − 1}e^{a+1}

and thus by induction the first two equalities are proved. The next two equalities follow by symmetry, by interchanging e with f and H_{1} with H_{2}. The last statement is an obvious special case of the equalities which precede it. The proof is complete. □

4.4 Lemma In the algebra B_{d} we have the identities

(a)  (H_{1} − H_{2})\binom{H_{2}}{b} = (d − 2b)\binom{H_{2}}{b} − (2b + 2)\binom{H_{2}}{b + 1}
(b)  (H_{2} − H_{1})\binom{H_{1}}{b} = (d − 2b)\binom{H_{1}}{b} − (2b + 2)\binom{H_{1}}{b + 1}

for any nonnegative integer b.

Proof. By 4.1(a) we have

(H_{1} − H_{2})\binom{H_{2}}{b} = (d − 2H_{2})\frac{H_{2}(H_{2} − 1) ⋯ (H_{2} − b + 1)}{b!}

= ((d − 2b) + (2b − 2H_{2}))\frac{H_{2}(H_{2} − 1) ⋯ (H_{2} − b + 1)}{b!}

= (d − 2b)\binom{H_{2}}{b} − 2\frac{H_{2}(H_{2} − 1) ⋯ (H_{2} − b + 1)(H_{2} − b)}{b!}

= (d − 2b)\binom{H_{2}}{b} − (2b + 2)\binom{H_{2}}{b + 1}

which proves the first identity. The second follows by symmetry. □

Recall the definition of height and degree for monomials given at the end of 3.4.

4.5 Theorem In the algebra B_{d} all monomials of the form e^{(a)}\binom{H_{1}}{b}f^{(c)}

(resp., f^{(a)}\binom{H_{2}}{b}e^{(c)}) of degree d+1 are expressible as ℚ-linear combinations of monomials of the same form but of strictly smaller degree and height.

Proof. By symmetry it is enough to prove the claim for monomials of the form e^{(a)}\binom{H_{1}}{b}f^{(c)}. For convenience, write M(a, b, c) for such a monomial.

The proof proceeds by induction on height. The truth of the claim for monomials of height zero is the content of 4.2(a).

Consider an arbitrary monomial M(a, b, c) of degree d + 1 and height s ≥ 1. We first consider the case a ≥ c. Then a must be larger than 0, and by an application of 3.5(a) and 3.4(a) we obtain the equalitites

e^{(a−1)}\binom{H_{1}}{b + 1}ef^{(c)} = e^{(a−1)}e\binom{H_{1} + 1}{b + 1}f^{(c)}

= ee^{(a−1)}\binom{H_{1}}{b + 1}f^{(c)} + ae^{(a)}\binom{H_{1}}{b}f^{(c)}.

By rearranging the above we find that

aM(a, b, c) = e^{(a−1)}\binom{H_{1}}{b + 1}ef^{(c)} − eM(a − 1, b + 1, c). (a)

By induction we know that M(a − 1, b + 1, c) is expressible as a ℚ-linear combination of monomials of degree ≤ d and height ≤ s−2, so eM(a−1, b+ 1, c) is expressible as a ℚ-linear combination of monomials of degree ≤ d + 1 and height ≤ s − 1. By induction (applied to the degree d + 1 terms) this reduces to a ℚ-linear combination of monomials of degree ≤ d and height ≤ s − 1.

Hence our claim for M(a, b, c) will follow once we show that the term

e^{(a−1)}\binom{H_{1}}{b + 1}ef^{(c)} can be expressed in the desired form. But, by 3.5(f) we

have

M(a − 1, b + 1, c)e = e^{(a−1)}\binom{H_{1}}{b + 1}f^{(c)}e

= e^{(a−1)}\binom{H_{1}}{b + 1} (ef^{(c)} + (H_{2} − H_{1} − c + 1)f^{(c−1)})

= e^{(a−1)}\binom{H_{1}}{b + 1}ef^{(c)} + e^{(a−1)}\binom{H_{1}}{b + 1}(H_{2} − H_{1} − c + 1)f^{(c−1)}.

By rearranging this we obtain the equality

\begin{aligned}e^{(a−1)} & \binom{H_{1}}{b + 1}ef^{(c)} = \\ & M(a − 1, b + 1, c)e − e^{(a−1)}\binom{H_{1}}{b + 1}(H_{2} − H_{1} − c + 1)f^{(c−1)}. \end{aligned} (b)

in which the second term on the right-hand-side is zero if c = 0, by the notational conventions introduced in 3.4. By Lemma 4.4 this second term is expressible as a linear combination of monomials of degree ≤ d+1 and height ≤ s − 2, and by induction it can be expressed as a ℚ-linear combination of monomials of degree ≤ d and height ≤ s − 2.

Finally, by induction we know that M(a − 1, b + 1, c) is expressible as a ℚ-linear combination of monomials of degree ≤ d + 1 and of height ≤ s − 2, and by induction applied to the terms of degree d + 1 this is reducible to a ℚ-linear combination of monomials of degree ≤ d and of height ≤ s−2. Thus it follows that M(a − 1, b + 1, c)e is expressible as a ℚ-linear combination of monomials of degree ≤ d + 1 and of height ≤ s − 1 (commuting the e to the front does not increase degree or height), so by applying induction one more time to the terms of degree d + 1 we are finished, in the case a ≥ c.

The argument in the other case c ≥ a is entirely similar and is omitted. □

5 Identifications

In this section we will prove that B_{d} is isomorphic with S(2, d). Theorems 2.1, 2.2, 2.3 will then follow.

5.1 Lemma Write \overline{X} for the image of X ∈ U(𝔤𝔩_{2}) under the representation ρ_{d} : U(𝔤𝔩_{2}) → End(E^{⊗d}). Then we have the following identities

\overline{H_{1}} + \overline{H_{2}} = d (a)

\overline{H_{1}}(\overline{H_{1}} − 1) ⋯ (\overline{H_{1}} − d) = 0 (b)

\overline{H_{2}}(\overline{H_{2}} − 1) ⋯ (\overline{H_{2}} − d) = 0 (c)

in the image S(2, d) of ρ_{d}. In fact, T (T − 1) … (T − d) is the minimal polynomial of \overline{H_{i}} for i = 1, 2.

Proof. Set W = E^{⊗(d−1)} so that E^{⊗d} ≃ E ⊗ W. We have the equality

ρ_{d}(H_{i}) = 1_{E} ⊗ ρ_{d−1}(H_{i}) + ρ(H_{i}) ⊗ 1_{W} .

Now clearly ρ(H_{1})+ρ(H_{2}) = ρ(H_{1} +H_{2}) = ρ(1) = 1. Moreover, the eigenvalues of ρ(H_{i}) are 0, 1 so its minimal polynomial is T (T − 1). Thus the results hold when d = 1.

For d > 1 we have by the above and induction that

ρ_{d}(H_{1}) + ρ_{d}(H_{2}) = 1_{E} ⊗ (ρ_{d−1}(H_{1}) + ρ_{d−1}(H_{2})) + (ρ(H_{1}) + ρ(H_{2})) ⊗ 1_{W}
= 1_{E} ⊗ (d − 1)1_{W} + 1_{E} ⊗ 1_{W}
= (d − 1)1_{E⊗W} + 1_{E⊗W} = d1_{E⊗W} .

This proves the first part. Moreover, by induction we may assume that the eigenvalues of ρ_{d−1}(H_{i}) are 0, 1, …, d − 1 (not counting multiplicities). Now suppose that f, g are diagonalizable linear operators on vector spaces E, W, resp. Then the eigenvalues of the linear operator 1 ⊗ g + f ⊗ 1 on E ⊗ W are all of the form λ + μ where λ is an eigenvalue of E and μ is an eigenvalue of W. Applying this fact to ρ_{d}(H_{i}) we see that its eigenvalues are 0, 1, …, d. The proof is complete. □

Remark. A similar argument shows also that the minimal polynomial of \overline{h} is (T + d)(T + d − 2) ⋯ (T − d + 2)(T − d).

5.2 Theorem Over ℚ, the Schur algebra S(2, d) is isomorphic with B_{d}. The set of all e^{a}H_{1}^{b}f^{c} such that a + b + c ≤ d is a ℚ-basis, as is the set of all f^{a}H_{2}^{b}e^{c} such that a + b + c ≤ d.

Proof. By the preceding lemma we see that the surjection ρ_{d} : U(𝔤𝔩_{2}) → S(2, d) factors through B_{d}, giving the commutative diagram

\begin{matrix}U(𝔤𝔩_{2}) & ⟶ & S(2, d) \\ ↓ & & ↑ \\ B_{d} & & B_{d} \end{matrix} (a)

in which all arrows are surjections. Since the set of all e^{a}H_{1}^{b_{1}}H_{2}^{b_{2}}f^{c} spans U(𝔤𝔩_{2}), it also spans B_{d}. But H_{2} = d − H_{1}, so it follows that the set of all e^{a}H_{1}^{b}f^{c} already spans. Now by Theorem 4.5 and it follows that the set of all e^{a}H_{1}^{b}f^{c} satisfying the constraint a + b + c ≤ d must span the algebra B_{d}. By a similar argument we see that the same statement holds for the set of all f^{a}H_{2}^{b}e^{c} such that a + b + c ≤ d.

In either case our spanning set is in one-to-one correspondence with the set of all monomials in 4 variables of total degree d (set one of the variables equal to 1 to get the correspondence). It is well known that this number is the dimension of S(2, d) (see [Gr]). Hence the map B_{d} → S(2, d) must be an isomorphism and the constrained spanning sets are bases. The theorem is proved. □

Remark. We will henceforth simplify the notation of Lemma 5.1, writing simply e, f, H_{1}, H_{2} for the images of these elements in S(2, d).

5.3 Theorem 2.2 follows as an immediate corollary of the above result, simply by using the relation H_{1} + H_{2} = d to eliminate H_{2} from the presentation of B_{d}. Similarly, by eliminating H_{1} we obtain Theorem 2.3.

We now prove Theorem 2.1. Start from Theorem 2.3 and set h = d − 2H_{2}. So H_{2} = (d − h)/2. By putting this into the relations 2.3(a), 2.3(b) and clearing denominators and multiplying the second relation by (−1)^{d} one arrives at the desired relations 2.1(a) and 2.1(b). Now the set of all f^{a}(h − d)^{b}e^{c} such that a + b + c ≤ d is a ℚ-basis for the algebra. But by expanding (h − d)^{b} by the binomial theorem we see that every such basis element is expressible as a ℚ-linear combination of the elements f^{a}h^{b}e^{c} satisfying a + b + c ≤ d. Thus this latter set spans the algebra. But it has the same cardinality as the dimension of the algebra, and so the proof is complete.

6 Integral reduction

The purpose of this section is to prove Theorem 2.4. For this we will need to know that the coefficients of the ℚ-linear combinations appearing in Theorem 4.5 are actually integers. This is the content of Theorem 2.5, which will be proved first.

Note that our proof does not rely on the results in the preceding section. Thus we could use the results of this section instead of Theorem 4.5 to give another proof of Theorems 2.1, 2.2, and 2.3.

6.1 Theorem We have in B_{d} the equalities

(a)  f^{(a)}\binom{H_{2}}{b}e^{(c)} = ∑_{k=1}^{min(a,c)}(−1)^{k−1}\binom{b + k}{k}f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k)}

(b)  e^{(a)}\binom{H_{1}}{b}f^{(c)} = ∑_{k=1}^{min(a,c)}(−1)^{k−1}\binom{b + k}{k}e^{(a−k)}\binom{H_{1}}{b + k}f^{(c−k)}

for all triples a, b, c of nonnegative integers satisfying the constraint a+b+c = d + 1.

Proof. This is by a double induction on a and c. The case a = c = 0 is 4.2(a) and the cases c = 0 (a, b arbitrary) and a = 0 (b, c arbitrary) are already given in Lemma 4.3.

By symmetry it suffices to prove just the first equality of the theorem, which can be rewritten in the form

0 = ∑_{k=0}^{min(a,c)}(−1)^{k}\binom{b + k}{k}f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k)} (c)

By induction this holds for some fixed triple a, b, c satisfying a + b + c = d + 1 and for all such triples whose first and last components are no larger than a and c. We show how to derive the result in the case a + 1, b − 1, c. The case a, b − 1, c + 1 is entirely similar and will be omitted.

The idea is to multiply (c) on the right by f and then to commute f all the way to the left. Using Lemma 3.5 we obtain

\begin{aligned}0 & = ∑_{k=0}^{min(a,c)}(−1)^{k}\binom{b + k}{k}f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k)}f \\ & = ∑_{k=0}^{min(a,c)}(−1)^{k}\binom{b + k}{k}f^{(a−k)}\binom{H_{2}}{b + k} (fe^{(c−k)} + Qe^{(c−k−1)}) \end{aligned}

where Q = H_{1} − H_{2} − (c − k) + 1. Then by Lemma 3.5 once again we arrive at the equality

\begin{aligned}0 & = ∑_{k=0}^{min(a,c)}(−1)^{k}\binom{b + k}{k}f^{(a−k)}f\binom{H_{2} + 1}{b + k}e^{(c−k)} \\ & + ∑_{k=0}^{min(a,c)}(−1)^{k}\binom{b + k}{k}f^{(a−k)}\binom{H_{2}}{b + k}(H_{1} − H_{2} − c + k + 1)e^{(c−k−1)}. \end{aligned} (d)

Now from Lemma 4.4 it follows that

(e) \binom{H_{2}}{b + k}(H_{1} − H_{2} − c + k + 1) = (a − k − b)\binom{H_{2}}{b + k}

− (2b + 2k + 2)\binom{H_{2}}{b + k + 1}

where we have made use of the equality (d − 2(b + k)) − (c − k − 1) = (d + 1 − b − c) − k − b = a − k − b.

Thus by putting (e) and the expansion

\binom{H_{2} + 1}{b + k} = \binom{H_{2}}{b + k} + \binom{H_{2}}{b + k − 1}

into (d) we obtain the equality

0 = f ∑_{k=0}^{min(a,c)}(−1)^{k}\binom{b + k}{k}f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k)}

+ ∑_{k=0}^{min(a,c)}(−1)^{k}\binom{b + k}{k}ff^{(a−k)}\binom{H_{2}}{b + k − 1}e^{(c−k)}

+ ∑_{k=0}^{min(a,c)}(−1)^{k}\binom{b + k}{k}(a − b − k)f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k−1)}

+ ∑_{k=0}^{min(a,c)}(−1)^{k}\binom{b + k}{k}(−2b − 2k − 2)f^{(a−k)}\binom{H_{2}}{b + k + 1}e^{(c−k−1)}.

(f)

Now by induction we claim that the first and fourth sums in the above equality are zero. This is clear for the first sum. To see this claim for the fourth sum, note that

\binom{b + k}{k}(−2b − 2k − 2) = −2(b + 1)\binom{b + k + 1}{k}

and so the fourth sum in (f) is equal to

−2(b + 1) ∑_{k=0}^{min(a,c)}(−1)^{k}\binom{b + k + 1}{k}f^{(a−k)}\binom{H_{2}}{b + k + 1}e^{(c−1−k)}

which is a multiple of the a, b + 1, c − 1 case of the theorem, and hence is zero by induction. (If min(a, c) = c then the last term in the sum is zero, so one can replace min(a, c) by min(a, c − 1) without changing the value of the sum.)

Thus the equality (f) reduces to

\begin{aligned}0 & = ∑_{k=0}^{min(a,c)}(−1)^{k}\binom{b + k}{k}(a + 1 − k)f^{(a+1−k)}\binom{H_{2}}{b + k − 1}e^{(c−k)} \\ & + ∑_{k=1}^{1+min(a,c)}(−1)^{k−1}\binom{b + k − 1}{k − 1}(a − b − k + 1)f^{(a+1−k)}\binom{H_{2}}{b + k − 1}e^{(c−k)} \end{aligned}

where we have shifted the index of summation in the second sum. Writing m for min(a, c) we thus obtain

\begin{aligned}0 & = ∑_{k=1}^{m}(−1)^{k}Rf^{(a+1−k)}\binom{H_{2}}{b − 1 + k}e^{(c−k)} \\ & + (a + 1)f^{(a+1)}\binom{H_{2}}{b − 1}e^{(c)} \\ & + (−1)^{m}\binom{b + m}{m}(a − b − m)f^{(a−m)}\binom{H_{2}}{b + m}e^{(c−1−m)} \end{aligned} (g)

where

R = \binom{b + k}{k}(a + 1 − k) − \binom{b + k − 1}{k − 1}(a − b − k + 1)

= (a + 1)\binom{b − 1 + k}{k}.

Note that the last term in (g) is zero if m = c. Otherwise m = a < c and the term takes the form

(−1)^{a}\binom{b + a}{a}(−b)\binom{H_{2}}{b + a}e^{(c−1−a)}

= (−1)^{a+1} (a + 1)\binom{b + a}{a + 1}\binom{H_{2}}{b + a}e^{(c−1−a)}

since

(−1)^{a}\binom{b + a}{a}(−b) = (−1)^{a+1}\binom{b + a}{b}b

= (−1)^{a+1}\binom{b + a}{b − 1}(a + 1)

= (−1)^{a+1} (a + 1)\binom{b + a}{a + 1}.

This shows that all the terms in equation (g) have a common factor of (a+1). Putting these terms together and dividing by (a + 1) we obtain the equality

0 = ∑_{k=0}^{M}(−1)^{k}\binom{b − 1 + k}{k}f^{(a+1−k)}\binom{H_{2}}{b − 1 + k}e^{(c−k)}

where M = m = min(a, c) = min(a + 1, c) in case m = c and M = m + 1 = a + 1 = min(a + 1, c) otherwise. In either case, M is the minimum of a + 1 and c, and so we have obtained at last the desired equality which completes the induction. □

6.2 Theorem Suppose that a, b, c are nonnegative integers such that s = a + b + c − d is positive. We have in B_{d} the equalities

(a) f^{(a)}\binom{H_{2}}{b}e^{(c)} = ∑_{k=s}^{min(a,c)}(−1)^{k−s}\binom{k − 1}{s − 1}\binom{b + k}{k}f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k)}

(b) e^{(a)}\binom{H_{1}}{b}f^{(c)} = ∑_{k=s}^{min(a,c)}(−1)^{k−s}\binom{k − 1}{s − 1}\binom{b + k}{k}e^{(a−k)}\binom{H_{1}}{b + k}f^{(c−k)}.

Proof. By symmetry it is enough to prove the first equality. We proceed by induction on s. The case s = 1 is precisely the content of Theorem 6.1.

Let a, b, c be given such that a + b + c − d = s + 1. If a < s then b + c − 1 ≥ d + 1, so by Lemma 4.3 we have:

either \binom{H_{2}}{b}e^{(c−1)} = 0 or \binom{H_{2}}{b − 1}e^{(c)} = 0.

In either case it follows that

f^{(a)}\binom{H_{2}}{b}e^{(c)} = 0.

By a similar argument one sees that this holds also if c < s. Hence we may assume that both a and c are ≥ s. It is enough to argue the result for the case a ≥ c ≥ s since the other case c ≥ a ≥ s is similar.

Thus a ≥ 1 and we have by the inductive hypothesis the equalities

f^{(a)}\binom{H_{2}}{b}e^{(c)} = \frac{f}{a} f^{(a−1)}\binom{H_{2}}{b}e^{(c)}

= \frac{f}{a} ∑_{k=s}^{min(a−1,c)}(−1)^{k−s}\binom{k − 1}{s − 1}\binom{b + k}{k}f^{(a−1−k)}\binom{H_{2}}{b + k}e^{(c−k)}

= \frac{1}{a} ∑_{k=s}^{min(a−1,c)}(−1)^{k−s}\binom{k − 1}{s − 1}\binom{b + k}{k}(a − k)f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k)}

= \frac{a − s}{a}\binom{b + s}{s}f^{(a−s)}\binom{H_{2}}{b + s}e^{(c−s)}

+ \frac{1}{a} ∑_{k=s+1}^{min(a−1,c)}(−1)^{k−s}\binom{k − 1}{s − 1}\binom{b + k}{k}(a − k)f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k)}.

Now the first term of the last equality above can be expanded by Theorem 6.1 (the base case of our present induction) since (a − s) + (b + s) + (c − s) = a + b + c − s = d + 1. Putting this in and shifting the index of summation, we obtain the equalities

f^{(a)}\binom{H_{2}}{b}e^{(c)}

= \frac{1}{a}\binom{b + s}{s}(a − s) ∑_{k=1}^{c−s}(−1)^{k−1}\binom{b + s + k}{k}f^{(a−s−k)}\binom{H_{2}}{b + s + k}e^{(c−s−k)}

− \frac{1}{a} ∑_{k=s+1}^{min(a−1,c)}(−1)^{k−(s+1)}\binom{k − 1}{s − 1}\binom{b + k}{k}(a − k)f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k)}

= \frac{1}{a}\binom{b + s}{s}(a − s) ∑_{k=s+1}^{c}(−1)^{k−(s+1)}\binom{b + k}{k − s}f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k)}

− \frac{1}{a} ∑_{k=s+1}^{min(a−1,c)}(−1)^{k−(s+1)}\binom{k − 1}{s − 1}\binom{b + k}{k}(a − k)f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k)}.

Now the second sum in the last equality above can be taken from s + 1 to c since min(a−1, c) is different from c only if a = c, in which case the additional term in the sum will be zero (the factor a−k is zero when k = c = a). Putting the two sums together and using the computation

\frac{a − s}{a}\binom{b + s}{s}\binom{b + k}{k − s} − \frac{a − k}{a} \binom{k − 1}{s − 1}\binom{b + k}{k}

= \frac{a − s}{a}\binom{k}{s}\binom{b + k}{k} − \frac{a − k}{a} \binom{k − 1}{s − 1}\binom{b + k}{k}

= \binom{k − 1}{s}\binom{b + k}{k}

we obtain that

f^{(a)}\binom{H_{2}}{b}e^{(c)} = ∑_{k=s+1}^{c}(−1)^{k−(s+1)}\binom{k − 1}{s}\binom{b + k}{k}f^{(a−k)}\binom{H_{2}}{b + k}e^{(c−k)}

and this completes the induction. □

The following combinatorial result is well-known (see [Ko]).

6.3 Lemma Let T_{1}, …, T_{m} be commuting indeterminates, and let F = F (T_{1}, …, T_{m}) be a polynomial over ℚ which takes on integer values whenever the variables are replaced by integers. Then F is expressible as an integral linear combination of the polynomials

\binom{T_{1}}{b_{1}}\binom{T_{2}}{b_{2}} ⋯ \binom{T_{m}}{b_{m}}

where b_{1}, …, b_{m} all belong to the set of non-negative integers and where b_{i} does not exceed the degree of F viewed as a polynomial in T_{i}.

6.4 Theorem 6.2 combines with the isomorphism B_{d} ≃ S(2, d) from Theorem 5.2 to prove Theorem 2.5.

We now prove Theorem 2.4. We begin with the representation

ρ_{d} : U(𝔤𝔩_{2}) → End(E^{⊗d}). (a)

We let U_{ℤ}(𝔤𝔩_{2}) denote the subring of U(𝔤𝔩_{2}) generated by all f^{(m)}, e^{(m)}, \binom{H_{1}}{m},

\binom{H_{2}}{m} for m ≥ 0. Then U_{ℤ}(𝔤𝔩_{2}) is the same as the algebra of distributions

on the ℤ-group G_{ℤ} = GL_{2} from 3.4.

Now the map in (a) restricts to give a map U_{ℤ}(𝔤𝔩_{2}) → End(E^{⊗d}). Let E_{ℤ} be the ℤ-submodule of E spanned by the canonical basis elements e_{1}, e_{2}. It is easy to see that E_{ℤ} is stable under the action of U_{ℤ}(𝔤𝔩_{2}). It follows that E_{ℤ}^{⊗d} is also stable, and so the image of the above map is contained within End(E_{ℤ}^{⊗d}). Thus we have a representation

ρ_{d}^{ℤ} : U_{ℤ}(𝔤𝔩_{2}) → End(E_{ℤ}^{⊗d}) (b)

By a result of Donkin [Do, 2.1] (see also [D]) the integral Schur algebra S_{ℤ}(2, d) is precisely the image of this representation.

Hence S_{ℤ}(2, d) is spanned over ℤ by (the image of) the elements

f^{(a)}\binom{H_{1}}{b_{1}}\binom{H_{2}}{b_{2}}e^{(c)}.

But by Lemma 6.3 we can express all \binom{H_{1}}{b_{1}}\binom{H_{2}}{b_{2}} = \binom{d − H_{2}}{b_{1}}\binom{H_{2}}{b_{2}} as

integral linear combinations of terms of the form \binom{H_{2}}{b}. Thus S_{ℤ}(2, d) is spanned by all f^{(a)}\binom{H_{2}}{b}e^{(c)}. But by Theorem 6.2 we see that the set of

such terms satisfying the restriction a + b + c ≤ d is an integral spanning set. But this set is linearly independent over ℚ, hence linearly independent over ℤ, hence an integral basis. This proves 2.4(a). Part (b) is proved similarly,

starting with the fact that the set of e^{(a)}\binom{H_{1}}{b_{1}}\binom{H_{2}}{b_{2}}f^{(c)} forms an integral

basis for U_{ℤ}(𝔤𝔩_{2}).

Since U_{ℤ}(𝔰𝔩_{2}) is generated by all divided powers e^{(m)}, f^{(m)} it is clear that U_{ℤ}(𝔰𝔩_{2}) is contained in U_{ℤ}(𝔤𝔩_{2}). It was shown in [Do] (see also [D]) that the image of U_{ℤ}(𝔰𝔩_{2}) in End(E_{ℤ}^{⊗d}) is the same as the image of U_{ℤ}(𝔤𝔩_{2}). This proves the first claim in Theorem 2.4. The proof is complete.

References

[Do] S. Donkin, On Schur algebras and related algebras III: integral representations, Math. Proc. Cambridge Philos. Soc. 116 (1994), 37–55.

[D] S. R. Doty, Polynomial representations of Chevalley groups, preprint, Loyola Univ. Chicago, 1999.

[DG] S. R. Doty and A. Giaquinto, Presenting quantum Schur algebras as quotients of the quantized universal enveloping algebra of 𝔤𝔩_{2}, preprint, Loyola Univ. Chicago, Sept. 2000.

[Gr] J. A. Green, Polynomial Representations of GL_{n}, (Lecture Notes in Math. 830), Springer-Verlag, New York 1980.

[RG] R. Green, q-Schur algebras as quotients of quantized enveloping algebras, J. Algebra 185 (1996), 660–687.

[Ja] J. C. Jantzen, Representations of Algebraic Groups, Academic Press, Orlando 1987.

[Ko] B. Kostant, Groups over ℤ, Proc. Symposia Pure Math. 9 (1966), 90–98.

Mathematical and Computer Sciences Loyola University Chicago Chicago, Illinois 60626 U.S.A. E-mail: doty@math.luc.edu

tonyg@math.luc.edu